Unit Circle Trigonometry: Foundations | SAT Math

[Video player would appear here]

Video introduction to the unit circle and trigonometry

Understanding the Unit Circle

The unit circle is a fundamental tool in trigonometry that relates angles to coordinates and trigonometric functions.

Key Features of the Unit Circle

[Diagram of unit circle with radius 1, showing x and y axes, and angle θ]

Circle with radius 1 centered at the origin (0,0)
Any point on the circle can be described as (cos θ, sin θ)
Angles measured from the positive x-axis (usually in radians)
One full rotation = 360° = 2π radians

Trigonometric Functions on the Unit Circle

Function	Definition	Relationship
Sine (sin θ)	y-coordinate	sin θ = y
Cosine (cos θ)	x-coordinate	cos θ = x
Tangent (tan θ)	y/x	tan θ = sin θ / cos θ

Quadrants and Signs

Quadrant	Angle Range	Signs (sin, cos, tan)
I	0 to π/2 (0° to 90°)	(+, +, +)
II	π/2 to π (90° to 180°)	(+, -, -)
III	π to 3π/2 (180° to 270°)	(-, -, +)
IV	3π/2 to 2π (270° to 360°)	(-, +, -)

Remember: All Students Take Calculus

All functions positive in Quadrant I
Sine positive in Quadrant II
Tangent positive in Quadrant III
Cosine positive in Quadrant IV

Special Angles and Reference Angles

[Diagram showing common angles (30°, 45°, 60°) and their coordinates]

Angle (θ)	Radians	sin θ	cos θ	tan θ
0°	0	0	1	0
30°	π/6	1/2	√3/2	√3/3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	undefined

Practice Question

If the terminal side of angle θ in standard position passes through the point (-3/5, 4/5) on the unit circle, what is the value of tan θ?

A) -3/4

B) -4/3

C) 3/4

D) 4/3

Explanation

On the unit circle:

x-coordinate = cos θ = -3/5
y-coordinate = sin θ = 4/5

The tangent of an angle is defined as:

tan θ = sin θ / cos θ = y-coordinate / x-coordinate

Substituting the given values:

tan θ = (4/5) / (-3/5) = (4/5) × (-5/3) = -4/3

The correct answer is B) -4/3.

Key observations:

The point (-3/5, 4/5) is in Quadrant II (negative x, positive y)
In Quadrant II, tangent values are negative (consistent with our result)
We simplified by multiplying by the reciprocal rather than dividing fractions

Common mistakes:

Choosing A) -3/4: Reversing x and y in the ratio
Choosing D) 4/3: Forgetting the negative sign from the x-coordinate
Not simplifying the fractions correctly

Additional Tips

Memorize the unit circle values for 0°, 30°, 45°, 60°, and 90° angles.
Remember that reference angles are always between 0° and 90°.
Use symmetry to find values for angles in other quadrants.
When converting between degrees and radians, remember π radians = 180°.
For angles not in the first quadrant, determine the sign based on the quadrant.